Statistical distribution of Fermat quotients

Victor Alexandru, Cristian Cobeli, Marian Vâjâitu, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


Let [Formula presented], for 0 ≤ b ≤ p2 − 1 and gcd(b, p) = 1, be the Fermat quotients arranged in the p × (p − 1) Fermat quotient matrix FQM(p). We study the elements of the matrix and prove two results: (i) Under the Generalized Riemann Hypothesis, ℤp is fully covered fast by the quotients qpq with q prime and q ≪ plog2p, as p → ∞ and (ii) The matrix FQMp) passes the pair size correlation test, in which any pair (u, v) of entries in the matrix, which lie apart under any apriori chosen fixed geometric pattern, are in the limit, as p → ∞, with equal probability 1/2 in the size relation u ≤ v or u ≥ v.

Original languageEnglish (US)
Article number112335
JournalChaos, solitons and fractals
StatePublished - Aug 2022


  • Fermat quotient matrix
  • Fermat quotients
  • Pair correlation size test
  • Wieferich primes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • General Mathematics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


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